We consider universal computability of the LCTRS with only rule scheme Calc:

  Signature: sumto :: Int -> Int -> Int

  Rules: sumto(x, y) -> 0 | x > y
         sumto(x, y) -> x + sumto(x + 1, y) | y >= x

The system is accessible function passing by a sort ordering that equates all sorts.

We start by computing the initial DP problem D1 = (P1, R UNION R_?, i, c), where:

  P1. (1) sumto#(x, y) => sumto#(x + 1, y) | y >= x

***** We apply the Integer Function Processor on D1 = (P1, R UNION R_?, i, c).

We use the following integer mapping:

  J(sumto#) = arg_2 - arg_1

We thus have:

  (1) y >= x |= y - x > y - (x + 1) (and y - x >= 0)

All DPs are strictly oriented, and may be removed.  Hence, this DP problem is finite.

Processor output: { }.